Consider a social network \(\mathcal {G} =(\mathcal {E})\) , where each node \(v_s \in  represents the user \) vs\( and each edge \) (vs, vt) E \( represents the friendship of two users \) vs\( and \) vt\(.Measuring the closeness of the friendship \) (vs,vt)\( is an important problem and finds numerous applications in real-world scenarios, where a user \) vs\( (called the source) \textit {only} interacts with the friend \) vt\( (called the target).For example, online gaming platforms often host friendship-enhancing events, which encourage the source \) vs\( to invite the target \) vt\( to return to the game or play together~\cite {luo2019efficient,luo2020efficient}. In this scenario, the friendship closeness measure is the backbone for understanding the invitation (resp. adoption) behavior of \) vs\( (resp. \) vt\(). Furthermore, this measure can be applied in recommending promising targets for the sources because the number of targets of a source could be considerable.Analogously, in the instant messaging platform WeChat, a source^{\prime }s post can only be liked and commented on by the target in contacts~\cite {zhang2018mobile}, thus the friendship closeness measure can also be leveraged to predict corresponding activities~\cite {zhang2021understanding,sankar2020inf,gomez2011uncovering}.\) In the present work, we focus on the topological friendship closeness (\({\mathtt {TFC}}\) ) measure that reflects the closeness of a source-target pair \((v_s,v_t)\)  on \(\mathcal {G} \)  rather than labels associated with them. In particular, most existing \({\mathtt {TFC}}\)  measures [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]} adopt the structural information between two users \(v_s\)  and \(v_t\) . For example, [5]} directly employs the tie strength of \((v_s,v_t)\) , which is measured by the times of historical interactions; [1]} proposes several measures in terms of the common neighborhood of \((v_s,v_t)\) . Furthermore, some measures preserve long and intricate paths between \((v_s,v_t)\)  based on random walks, e.g., personalized PageRank [2]}.
Unfortunately, the above-said measures may render compromised results, as they ignore the group information and the potential group effect related to \((v_s,v_t)\) , which says a user's belonging group can influence his/her decision.
